The Fractional Fourier Transform (FrFT) has a wide range of applications in fields such as optics, quantum mechanics, image processing, data compression, and signal processing for communications. The FrFT of a function ƒ(x) of order a is defined asFa[ƒ(x)]=∫−∞∞Ba(x,x′)ƒ(x′)dx′  (1)
where the kernel Ba(x,x′) is defined as
                                          B            a                    ⁡                      (                          x              ,                              x                ′                                      )                          =                                            e                              i                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                                                                        ϕ                          ^                                                /                        4                                                              -                                          ϕ                      /                      2                                                        )                                                                                                                      sin                  ⁡                                      (                    ϕ                    )                                                                                              1                /                2                                              ×                      e                          i              ⁢                                                          ⁢                              π                ⁡                                  (                                                            x                      2                                        -                                          cot                      ⁡                                              (                        ϕ                        )                                                              -                                          2                      ⁢                                                                                          ⁢                                              xx                        ′                                            ⁢                                              csc                        ⁡                                                  (                          ϕ                          )                                                                                      +                                                                  x                                                  ′                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              cot                        ⁡                                                  (                          ϕ                          )                                                                                                      )                                                                                        (        2        )            
where ϕ=aπ/2 and {circumflex over (ϕ)}=sgn[sin(ϕ)]. This applies to the range 0<|ϕ|<π, or 0<|a|<2. In discrete time, the N×1 FrFT of an N×1 vector can be modeled asXa=Fax  (3)
where Fa is an N×N matrix whose elements are given by
                                          F            a                    ⁡                      [                          m              ,              n                        ]                          =                              ∑                                          k                =                0                            ,                              k                ≠                                  (                                      N                    -                    1                    +                                                                  (                        N                        )                                                                    2                        )                                                                              )                                                      N                    ⁢                                                    u                k                            ⁡                              [                m                ]                                      ⁢                          e                                                -                  j                                ⁢                                  π                  2                                ⁢                ka                                      ⁢                                          u                k                            ⁡                              [                n                ]                                                                        (        4        )            
and where uk[m] and uk[n] are the eigenvectors of the matrix S defined by
                    S        =                  (                                                                      C                  0                                                            1                                            0                                            …                                            1                                                                    1                                                              C                  1                                                            1                                            …                                            0                                                                    0                                            1                                                              C                  2                                                            …                                            0                                                                    ⋮                                            ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    1                                            0                                            0                                            …                                                              C                                      N                    -                    1                                                                                )                                    (        5        )            
and
                              C          n                =                              2            ⁢                                                  ⁢                          cos              ⁡                              (                                                                            2                      ⁢                      π                                        N                                    ⁢                  n                                )                                              -          4                                    (        6        )            
In Eq. (6), N is the size of the matrix determined by the length of the vector x. The FrFT is a useful approach for separating a signal-of-interest (SOI) from interference and/or noise when the statistics of either are non-stationary (i.e., at least one device is moving, Doppler shift occurs, time-varying signals exist, there are drifting frequencies, etc.). The FrFT enables translation of the received signal to an axis in the time-frequency plane where the SOI and interference/noise are not separable in the frequency domain (as produced by a conventional Fast Fourier Transform (FFT)) or in the time domain
The Wigner Distribution (WD) is a time-frequency representation of a signal. The WD may be viewed as a generalization of the Fourier Transform, which is solely the frequency representation. The WD of a signal x(t) can be written asWx(t,ƒ)=∫−∞∞x(t+τ/2)x*(t−τ/2)e−2πjτƒdτ  (7)
The projection of the WD of a signal x(t) onto an axis ta gives the energy of the signal in the FrFT domain a, |Xa(t)|2. Letting α=aπ/2, this may be written as|Xα(t)|2=∫−∞∞Wx(t cos(α)−ƒ sin(α),t sin(α)+ƒ cos(α))df  (8)
In discrete time, the WD of a signal x[n] can be written as
                                          W            x                    ⁡                      [                                          n                                  2                  ⁢                                      f                    s                                                              ,                                                kf                  s                                                  2                  ⁢                  N                                                      ]                          =                              e                          j              ⁢                              π                N                            ⁢              kn                                ⁢                                    ∑                              m                =                                  l                  1                                                            l                2                                      ⁢                                          x                ⁡                                  [                  m                  ]                                            ⁢                                                x                  *                                ⁡                                  [                                      n                    -                    m                                    ]                                            ⁢                              e                                  j                  ⁢                                                            2                      ⁢                      π                                        N                                    ⁢                  k                  ⁢                                                                          ⁢                  m                                                                                        (        9        )            
where l1=max(0, n−(N−1)) and l2=min(n, N−1). This particular implementation of the discrete WD is valid for non-periodic signals. Aliasing is avoided by oversampling the signal x[n] using a sampling rate ƒs. (samples per second) that is at least twice the Nyquist rate.
When applying the FrFT to perform interference suppression, the rotational parameter a should first be estimated. Radar echoes from moving targets are chirp signals, and the WD of a chirp signal x(t) is shown in graph 100 of FIG. 1. Graph 100 illustrates how the FrFT may be used to detect chirp signals. By rotating to the axis ta and computing the energy in the FrFT, the chirp projects onto the axis as a strong tone, which can be detected and notched by finding the peak of the energy using Eq. (8). Lower power chirps, at other rotational axes, can then be detected by repeating the process.
The signal model discussed here follows that presented by Sun et al. See Sun et al., Application of the Fractional Fourier Transform to Moving Target Detection in Airborne SAR, IEEE Trans. on Aerosp. and Electr. Systems, Vol. 38, No. 4 (October 2002). Assume that there is an airborne radar platform, such as an aircraft moving along the y-axis at constant speed V and a moving target i at an initial distance R0,i at time t=0, moving with speed vi and acceleration ai, potentially in a different direction than the plane. See, e.g., graph 200 of FIG. 2. After time t, the platform and target have moved a total distance of Vt and vit+½ait2, respectively. The components of the speed of the target along the x-axis and y-axis are vr,i and vc,i, respectively. Similarly, the accelerations are ar,i and ac,i, respectively, where c is the speed of light at approximately 3×108 m/s. At time t, target i is now at a distance from the platform whose horizontal component has decreased by vr,it+½ar,it2. Likewise, the vertical component has decreased by vc,it+½ac,it2. Accordingly, the distance between the target and radar can be approximated by
                                          R            i                    ⁡                      (            t            )                          ≈                              R                          0              ,              i                                -                                    v                              r                ,                i                                      ⁢            t                    +                                    [                                                                    (                                          V                      -                                              v                                                  c                          ,                          i                                                                                      )                                    2                                -                                                      R                                          0                      ,                      i                                                        ⁢                                      a                                          r                      ,                      i                                                                                  ]                        ⁢                                          t                2                                            2                ⁢                                  R                                      0                    ,                    1                                                                                                          (        10        )            
The echo of the moving target as received by the radar system can be approximated by a chirp signal that takes the formxi(t)=ej2πƒd,itejπKit2  (11)
where
                              f                      d            ,            i                          =                              2            ⁢                          v                              r                ,                i                                              λ                                    (        12        )                                          K          i                =                              2                          λ              ⁢                                                          ⁢                              R                                  0                  ,                  i                                                              ⁡                      [                                          -                                                      (                                          V                      -                                              v                                                  c                          ,                          i                                                                                      )                                    2                                            +                                                R                                      0                    ,                    i                                                  ⁢                                  a                                      r                    ,                    i                                                                        ]                                              (        13        )            
λ is the wavelength of the radar, related to its frequency ƒ by λ=c/ƒ. It is further assumed that there are no strong point scatterers so the ground (i.e., surface) clutter can be adequately modeled as additive white Gaussian noise (AWGN) using a desired Signal-to-Clutter Ratio (SCR). If a scenario with multiple moving targets (K total) is assumed, the composite received echo signal can be written as
                              x          ⁡                      (            t            )                          =                                            ∑                              i                =                1                            K                        ⁢                                          A                i                            ⁢                                                x                  i                                ⁡                                  (                  t                  )                                                              +                      n            ⁡                          (              t              )                                                          (        14        )            
where n(t) is a combination of clutter and noise using a given SCR and each amplitude Ai is chosen to model strong and weak targets. If it is assumed, without loss of generality, that signal i=1 is the main target, the amplitudes Ai for i=1, 2, . . . , K can be modeled using an assumed carrier-to-interference (CIR) by writing Ai=10−CIRi/20.
The algorithm given by Sun et al. is based on the fact that a chirp signal in the FFT domain is spread out over a band of frequencies, whereas at the optimum rotational parameter a (see FIG. 1), the projection of the signal onto that axis, given by the magnitude squared of its FrFT, is maximized. This signal is therefore estimated by computing a peak and comparing it to the sidelobe level to determine a detection (i.e., peak-to-sidelobe ratios (PSRs)). The peak is then removed by applying a narrow, bandstop filter. Next, by rotating back to the time domain, the operation is performed again to find the second strongest signal, the third strongest, etc.
It should be noted that Sun et al. never found a tone. Rather, the peak was calculated at multiple values of a in the FrFT domain. The PSR was then computed and the value of a that gave the largest PSR was chosen.
Per the above, the FrFT can be applied to the problem of separating multiple moving targets of differing power levels received by a monostatic radar system in clutter, for instance, because moving target echoes are chirp signals, which can be separated in the FrFT domain. However, conventional processes, such as that presented in Sun et al., are relatively computationally intensive and not presently capable of performance in real time. Accordingly, an improved approach to separating weak and strong moving targets may be beneficial.